14. Sequences & Series

9–15. Geometric sums Evaluate each geometric sum.

∑ k = 0 to 83ᵏ

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The sum ∑ (k = 1 to ∞) 1 / 3ᵏ is a p-series.

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

{Use of Tech} Error in a finite alternating sum

How many terms of the series ∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k⁴ must be summed to ensure that the approximation error is less than 10⁻⁸?

Define the remainder of an infinite series.

21–42. Geometric series Evaluate each geometric series or state that it diverges.  

21.∑ (k = 0 to ∞) (1/4)ᵏ

39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) for the exact value of the series.

39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

Find the first term a and the ratio r of each geometric series.

a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))

Geometric sums

Evaluate the geometric sums

∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.

a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.

∑ (k = 0 to ∞) (–2/7)ᵏ

67–70. Formulas for sequences of partial sums Consider the following infinite series.

b.Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.

∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.

b. Evaluate the series using Theorem 10.7.

∑ (k = 0 to ∞) (–2/7)ᵏ

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 2 to ∞) √k / (ln¹⁰ k)